A Symmetric Functional Calculus for Systems of Operators of Type ω

For a system A = (A _i ,…, A _n ) of linear operators whose real linear combinations have spectra contained in a fixed sector in ℂ and satisfy resolvent bounds there, functions f(A) of the system A of operators can be formed for monogenic functions f having decay at zero and infinity in a corresponding sector in ℝn+1. In the case that the operators A _i ,…, A _n commute with each other and satisfy square function estimates in Hilbert space, the correspondence between bounded monogenic functions defined in a sector in ℝn+1 and bounded holomorphic functions defined in a sector in ℂn is used to define the functional calculus f→f(A) for bounded holomorphic functions f in a sector of ℂn. The treatment includes the Dirac operator on a Lipschitz surface in ℝn+1.